# Full text of "Treatise On Analysis Vol-Ii"

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```3    SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA       327

have Di(/« p,) <; ( f/2"" d^Y" for all n 2> 1, and use Problem 12(e) of Section 13.11.

Use this problem also to show that (y) implies (a).)

(c) Let p ^> 1 and let p, be a positive measure on X. For each measure 9 on X, let
dd/dp, denote the function h (defined up to equivalence) which features in the Lebesgue
decomposition (13.18.4) 6 — h • ft + or, where |a| is disjoint from p.. Show that every
representative measure for x has base p, if and only if

for every positive measure v on X. (To show that the condition is necessary, reduce
to the case p > 1 and Dp(v) > 0. Then apply Problem 10(d) and Holder's inequality.)
(d) Let;? ^> 1 and let p be a positive measure on X with total mass 1. Then the follow-
ing conditions are equivalent:

(a)   p, is the unique representative measure for x-

(/3)    Dp(v) = J^dv/dp,) for all positive measures v on X.

(y)   Dp(v) <^ J^dv/dp) for all positive measures v on X.

(S)   \\d6 ^ JM( | <#?/d]u, |) for all quasi-representative measures 0 for %. ("Szego-

Kolmogoroff-Krein theorem").

(To prove that (y) implies (a), use (c) to show that p, is hyperextremal, and then
use (b).)

12. With the hypotheses and notation of Problem 8, let /x be a representative measure
for x- Let K.(p) denote the set of functions fe £>1(p) such that /• //, is a quasi-
representative measure for x» in other words such that ufdp — ( /d/zll (u dp.\

- ft) - W|). We have B c K(JLI).

for all u e B. Then

(a)    For each function g^.0 in -^(/x), show that there exists a function fe K(ft)
such that |/| <;# and j fdp, = DI(|/| • p) = D^g - p) (use Problem 10(d)).

(b)    A function /e K(/x) is said to be exterior if \\fdp = Di(|/| • p) >0, and

interior if |/(x)| = 1 almost everywhere with respect to p,. Let E(/x) (resp. I(/i)) be
the set of exterior (resp. interior) functions. The intersection E(JU-) n I(/x) consists of
functions equivalent to constants c such that |c| = 1.

(c)    Let g 1> 0 be a function in ^l(p) such that Dj(# • /x) « J^g) > 0. Show that
there exists a function /e E(p) such that |/| ~<^. (Use (a) by writing/^^/z, where
h e JS^Ou) and JM(|/r|) = 1, and use Problem 12(e) of Section 13.11.)

13. With the hypotheses and notation of Problem 8, let p, be a hyperextremal represen-
tative measure for x- Then DI(/- p.) == J^(/) for all/ ^ 0 in &l(p) (Problem ll(b)).
(a) Let/e K(/x). For/to be an exterior function (Problem 12), it is necessary and

sufficient that inf I 11 — uf\ du — 0. (To show that this condition is sufficient, prove

ueB J

that it implies that there exists a sequence (wn) in B such that x(w/») ~ 1 fc>r all n an<^

. To show that the condition is necessary, we

such that    \unf\ dp, tends to   \fd\t. • /x). Conclude that, for all functions /g> 0 in ^(ft), we).n particular, the image of C under the
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